Bernoulli convolutions and an intermediate value theorem for entropies of K-partitions

We establish a strong regularity property for the distributions of the random sums ∑ ± λⁿ, known as "infinite Bernoulli convolutions": For a.e. λ ∈ (1/2, 1) and any fixed l, the conditional distribution of (ω_n+1, ω_n+l) given the sum ∑_n=0 ^∞= ω_nλⁿ, tends to the uniform distribution on (±1)^l as n → ∞. More precise results, where l grows linearly in n, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropy h has K-partitions of any prescribed conditional entropy in [0, h]. This answers a question of Rokhlin and Sinai from the 1960's, for the case of Bernoulli systems.